No one knows for sure, but likely group:

Juhani Tapiola and Seppo Tiainen, and Matti Makkonen . The first inklings came as early as 1982.
They worked for FMK (Framtida mobila kommunikationer, or Future Mobile Communications), a group which was researching GSM..

Most places I saw online list the inventor as Matti Makkonen, but as you can see in this article, he rejects the idea that it was his invention.

A direct quote from referenced source:

There are numerous explanations for how text messaging was developed.

One involves early paging systems such as MBS (introduced in Sweden in 1978) and Minicall (introduced in 1985). Anyone with MBS could use their own phone to send a message to someone else who had a portable receiver, says Östen Mäkitalo.

Like MBS, Minicall also offered one-way transmission of groups of digits and text messages: they were sent from the base station without the mobile being able to acknowledge reception. For marketing reasons the text function was not implemented in Minicall until 1988.

The embryo of text messaging could also be found in NMT, “a forerunner for a simple data service of the SMS type, laughingly referred to as ‘poor man’s data’”, says Thomas Beijer. Thomas Haug describes how, during the work on specifications for NMT 450, there were also ideas “about linking fax machines to mobiles, which it was felt could be practical for journalists in the field, for instance, who needed to communicate with news desks”.

Beijer adds: “This is one of the favorite talking points in the history of radio. Over the years there have been hundreds of ideas about how to send text messages using radio. Examples include maritime telex (MARITEX) and Inmarsat. I think you could say that whenever a new radio system has been invented there have been more or less fruitful discussions about how it could be used to transmit text. My conclusion is that as a concept the idea is almost a century old. The real challenge was to take it from the conceptual level to an application that works technologically and commercially.”
‘Text clicker’

One account that has received widespread circulation is that three Finnish engineers invented text messaging over a few beers in a pizzeria in Copenhagen one summer evening in 1982. Matti Makkonen, who played an active role in both the NMT process and the meetings of the GSM group in 1982 and 1983, gives the following account:

The NMT group had set up a separate Nordic working group, FMK (Framtida mobila kommunikationer, or Future Mobile Communications) to study the possibilities of a digital mobile system. This group met in four nordic countries between 1982 and 1985 and then continued its work in the corresponding GSM working groups.

Makkonen had arrived together with Juhani Tapiola and Seppo Tiainen, his colleagues from the Finnish telecommunications agency, the evening before the FMK meeting, and they were sitting in a pizzeria preparing for it. “The question we asked ourselves was what practical applications could be interesting in a digital system,” Makkonen says.

A starting point was the paging system that had recently become popular. A small display allowed the subscriber to see the number of the caller trying to contact them. In principle, it could include text as well. “But the only way to send text was to ring the operator first and ask for the message to be sent using the operator’s equipment.”

“We sat there discussing the route taken by a text message from a mobile through the paging system. Juhani had brought a programmable calculator with him. We realized that the numerical keypad could also be used for letters. Then it struck us that the paging system was not needed, it was just a detour. There was nothing to stop the mobile phone itself from receiving a text message.”

The three then began lightheartedly to make a list of different uses text messages could be put to. They could see that messages could be sent and answered without anyone else being disturbed and that messages could be read and answered when it suited the recipient. They invented a Finnish word for their innovation: “tekstinäpellin” (roughly “text clicker”).

It is unclear what happened next to Juhani’s, Seppo’s and Matti’s idea.

SMS

It is clear that the Short Message Service (SMS) used extensively today all around the world was developed during the work on GSM.

Two basic ideas were presented. The Nordic representatives proposed a system to deal with messages based on what was known as the X.400 protocol, an address format for e-mail. The Franco-German representatives proposed a method that would use the signaling channel in the mobile system.

This latter suggestion had its roots in a Franco-German cooperative project in 1983–84 that led to the extended Franco-German involvement in GSM work. The proposal was first presented to the GSM group at a meeting in Oslo in February 1985.

The Franco-German concept was then processed in the GSM organization and its technical specifications laid down in a small working group that started in 1987. The first chairman of this group was a Norwegian, Finn Trosby, who said in 2009: “There is no individual or company that can claim to be the ‘father’ or ‘creator’ of any service or important function produced in the course of the development work that went into GSM. The GSM project was multinational cooperation at its best.”

Matti Makkonen has been referred to in different contexts as the “father of text messaging” but he rejects this epithet. “The SMS function is the result of extensive and open international cooperation, and GSM documents prove that it is based on the Franco-German proposal,” he says.

Seppo Tianien adds that in the 1990s, SMS was also specified for NMT and was used for instance in Poland and Russia, but not for NMT in the Scandinavian countries. “Nokia’s range included small handheld NMT mobiles at the end of the 1990s. From the user’s point of view they dealt with SMS in the same way as GSM telephones.”

Matti Makonnen, a very modest guy from Finland who changed millions of lives. He was paid about $500 for this job by Sonera, and now he's one of VPs in Nokia.