There are future programmers in the space epic A Deepness In The Sky and they are not described with any fondness.

Bullshit, we’ve just never (yet) seen Trek Support: The Movie.

Starfleet officers often perform tasks that we would called programming. Most often its engineers doing so (a term we use for programmers now!)

B'elanna Torres reprograms a Cardassian ship's computer. The Doctor is a holographic program coded by a Starfleet researcher. Multiple ship's crew across shows and seasons code their own holodeck 'programs'. Chief O'Brien makes computer subsystems to house a sentient computer program. Spock often dealt with computers and analysed them in ways akin to software analysis, including breaking a program by feeding it paradoxical information.

Star Trek may not call anyone a programmer by name, but clearly the science and engineering officers are proficient programmers to varying degrees.