A relevant comparison, then, might be Gunsmoke, which was a tetrad instead of a triad. Kitty as a business owner and important part of the local economy was an independent woman and equal, more or less, to Matt and Doc. Matt and Kitty are close in age, have similar "on the frontier" experiences and their friendship is based on mutual respect and not, except for rare hints, romantic. The problem may be the shipboard setting. If the first officer and ship's doctor slots are filled, it's hard to think of a position which could be on a near-equal footing with the captain. Another doctor, a psychologist like Dr Dehner? A female chief engineer?
The casting would be important, too; I don't know if Marianna Hill or GLW would be right for a non-romantic co-star with Shatner. But Diana Muldaur, I can see that.