I guess I don't have much authority to state my favorite. I only played a bit of Breath of Fire 1 and only didn't beat the final boss in Breath of Fire 3.
Basically I have to say Breath of Fire 3 is my favorite since it is the only one I truly played. I love many things about Breath of Fire 3 although I think the art was a mixed bag actually. This was the first game to ever grab me by its story first (I was bad at understanding stories when I was younger, but this game was very different in its storytelling). I think the dragon gene system was just amazing and it was good for equalizing the difficulty of the game (although I still remember the game being difficult). Becoming a dragon also was good enough to compete with the existence of limit breaks in that time period, as I played every game expecting them to have a limit break system, usually only to be disappointed. I also loved the mentor system, I just wish it wasn't so time consuming to do (although it wouldn't be if you knew the level up trick). The fairy village is an amazing concept as well, although I forget exactly how it works. It amazed me that they had the fairy village when it was so mysterious. Also, the fishing minigame was pretty fun. Even though I hated the type of music that Breath of Fire 3 uses, the way they used it really sold me. That boss battle theme is so atypical that it made me fall in love with it.
In Breath of Fire 1, I felt like it was a pretty solid game, but going through the dungeons was a pain and I was wishing for dragon form like in 3. The start of the game is difficult and grindy if I remember correctly. The encounters on the world map were very cool and the story felt nice so far, with introducing the female positive themes that seem to also exist in 2, but I don't know anything about 4 (to avoid spoiling it for myself). Dragon Quarter looks like a game I would enjoy. I watched a lot of gameplay from a longplay or something, but I skipped the story scenes. I really like the atmosphere of the game and the very pressured design/difficulty. It works more as a horror/thriller than the others, but I think it was a much better attempt to change things up than 6 and it didn't feel too out of sync with what I knew about Breath of Fire.