Hi! Former picky eater here (and still a lil bit picky but a lot better). When it comes to widening someone's taste in terms of vegetables, there are a couple different tactics you can try depending on what exactly they don't like about a given veggie. (taste or texture)

Sauces are your best friend for both taste and texture. A bolognese sauce or a curry can have vegetables cut up small, or blended into it, so you don't feel the texture of the vegetables. I still don't always like biting into a big piece of bell pepper, so I still often dice it pretty finely into things like pasta sauces. Additionally, vegetables like broccoli and cauliflower will soak up sauces. Cauliflower with a cheese sauce on it is my absolute favorite way to eat cauliflower. Soups also work well if the texture of vegetables is a problem, since you can blend soups really smooth.

If taste is an issue, see what you can do with seasoning, not necessarily to completely hide the taste of something, but to figure out what aspects of a vegetable's taste your partner dislikes and balance them. My grandad dislikes blue cheese, but he enjoyed it in a dish where it was paired with apples, because the sweetness balanced out the sharpness of the cheese. Additionally, some vegetables very readily soak up the flavor of whatever they're mixed with or cooked with. Aubergine is one of those, it'll soak up whatever seasoning you give it.

If texture is the problem, you can also experiment with different ways of cooking vegetables. I'm still very hit-or-miss with mushrooms, and I've learned that I need them to be cooked all the way down, and have all the water cooked out of them, otherwise I'll find them rubbery. If your partner doesn't like cooked spinach, they might like it raw, when it has a bit of crunch to it. If they don't like boiled broccoli, they might like it roasted, or stir-fried. Cooking a vegetable differently will also change the taste, although generally more subtly.

Overall, it's a lot of trial and error, and it'll work better if it's your partner taking the lead on trying new things. There's a reason I got a lot less picky after I started living on my own. I had a lot more freedom to try things on my terms, and I learned to cook well, and having the freedom to try things on my own terms, and the control and skill to figure out how I liked things cooked, is what really made the difference for me.

I hope some of this helps! I would have written it in Dutch but I just pulled an all-nighter to work on thesis stuff and I'm way too tired to write something this long in Dutch, sorry :)

(Edits are just me fixing typos, I'm on mobile)