Yes, you can catch the shiny version of Turtonator in Pokemon GO, but only if you are very lucky.
Any Turtonator you encounter in Pokemon GO has a very low chance of being shiny. However, it is also difficult to encounter at the moment.
Currently, Turtonator is only available to catch after defeating it in Tier 3 Raids, but, even then, it is only a Raid boss from June 29 to July 2, 2023.
Therefore, to get a shiny Turtonator when it comes to the game, you will need to battle in loads of Turtonator Raids in this short period of time, and hope you are lucky. Alternatively, you could just wait a while until it becomes a wild spawn at a later date.

How to Get Shiny Turtonator in Pokemon GO
As the only way to encounter Turtonator in Pokemon GO is by defeating it in a Tier 3 Raid, this is also the only way to get a shiny Turtonator.
The exact shiny odds for Turtonator have not been determined yet. However, we expect that it will be 0.8%, or 1 in 125.
This is a slightly boosted shiny rate for rare Pokemon that do not spawn very often. The standard shiny rate for Pokemon is 1 in 500.

The shiny version of Turtonator’s shell and the top of its head is brown and yellow. Meanwhile, its feet and tail are brown, and its body is a blue-green color.
This makes it appear very different to the light green standard Turtonator that has a red and yellow instead of brown and yellow details.