There are some homebrew variants here on BGG for solo play, though I have not tried any of them. You could try the Borg cooperative scenario solo, but it's quite difficult with fewer players. It might be a hassle, but you could play that scenario using multiple decks (as if you were multiple players). Or you could play it with just one deck, but give yourself one of the stronger starships to start the game.
If your wife likes the Original Series, there's a new standalone set for TOS coming out very soon.
I've been playing solo for a couple of weeks. I use the standard rules, with a few exceptions: -a)Only playable with the standard Exploration scenario. Forget about the Klingon Civil War and Reunification Scenarios, since they require teams to play. And I tried one of the Borg scenarios. Got assimilated after 8 or 9 turns. -b)When I flip a "War" Event, I put it aside and flip a new Space card. Solo play has the advantage of familiarizing you with the rules so that when you play with friends, everything goes faster.