The British No. 2 cruised through his opening round of qualifying on Wednesday, beating Liang-Chi Huang 6-1 6-2, but was dumped out by Hungarian Fucsovics.
Birmingham-born Evans struggled to find any rhythm in his second-round tie, despite battling to 4-4 in the first set. Evans appeared to lose his concentration from there on in, as former junior Wimbledon champion Fucsovics reeled off seven games in a row en route to a comfortable victory.
Watson, who managed to reach the third round of the season-opening Grand Slam last year, defeated Australian Arina Rodionova 6-4 6-3 and will meet Frenchwoman Stephanie Foretz Gacon in the next round.
After playing her first match in Grand Slam qualification in 18 months following illness, Watson said: "I was very nervous. I feel like in the middle I loosened up but my energy really dropped at the end because I think all the nervous energy had worn me out.
"But I was able to close it out. I played quite well, I definitely can play better, and I think I'll play a lot freer in my next match.
"I don't know why I was so nervous, it's a while since I played in qualifying and I expect myself to do well and go far. I really like the tournament and play well on these courts so I think I'm putting too much pressure on myself."
Konta also breezed through her opening round match, beating American Grace Min 6-2 6-1, and will now face the winner of the match between Sachie Ishizu and Olga Savchuk.
British No. 4 Tara Moore suffered defeat in qualification, losing 6-0 7-5 to Japan’s Erika Sema.