At how many different scales were starship Enterprise models produced?

Production models

It looks like five different scales were used for models used in production and shown on screen.

A 33" model was produced as a pre-production reference model, and for publicity shots. This model did make it onto TV, though, as "stock footage", part of the intro, and also as a desktop model in one episode. (The production notes say this is about 1:192 - but that scale was based on an original design of only 540' for the Enterprise. The final design had the Enterprise had 947', so this scale is actually 1:344)

An 11' model (your 1:4 scaling there (11' = 4*33")) was used for actual production. Only the one model was made, but it was heavily retrofitted once the series was picked up after the pilot to allow internal lighting (instead of having painted on windows).

A 4" (1:2841) model was produced, but it doesn't look like that one made it onto the screen. Two 3" (1:3788) metal models were also produced in season 2, primarily for reference purposes.

Finally, some 1:650 licenced model kits were actually used in production (in particular, as a battle-distressed NCC-1071 Constellation, or as models on board the Enterprise in the movies).

Licenced model kits

At least eleven scales were used by various kit producers. These scales do not include a multitude of toys/trinkets that have been produced.

There have been a lot of Star Trek model kits produced over the years - and the Enterprise models have been scaled at 1:2500, 1:2200, 1:1400, 1:1000, 1:850, 1:700, 1:650, 1:600, 1:537, 1:500, and 1:350, by various producers over the years.

Why 1:350 for the Polar Lights model? 1:350 has become a standard size for navy models over the last 40 years - so tooling may be the determining factor here - but also note, this is the biggest scale used so far, which means a lot more detailing for the fans.

It seems that 1:1400 is also used for a lot of the current Star Trek models (there's your 1/4 factor, again).