World number 111 Vekic won her first Grand Slam match with a 6-1 6-2 demolition of Czech Andrea Hlavackova, ranked 45 places above her, to reach round two.
Coached by Englishman David Felgate and with a training base in London, the temptation for British authorities to ask her if she would like to switch will be high.
But Vekic said there was no chance she would be swapping the Croatian flag for a British one.
"No, I'm very happy playing for Croatia," she said. "Us Croatians have big Croatian hearts so we go out there fighting on the court, fighting for our country."
Vekic reached her first WTA Tour final last year and showed more evidence that she is a rising talent of the women's game with an assured performance.
"It was amazing," Vekic said. "I was quite nervous at the beginning but as the match went on, I started playing my tennis and it was really good.
"I definitely didn't expect it to be that easy, she's a great player and she played better in the second set but I can't really complain with that win."
Her reward for beating Hlavackova is a meeting with former world number one Caroline Wozniacki and Vekic said she cannot wait to test herself.
"Obviously she was number one for quite a while and I just want to go out there and see how it is to play against a player like that," she said. "I just want to see if I can do any damage and see how it goes."
Felgate, who coached former world number four Tim Henman for most of his career, began working with Vekic when she was 11 and when the Croat is not at school in Osijek or playing tournaments, she trains in London, accompanied by her mother.
Felgate said Vekic's attitude is second to none and was delighted with her performance on such a big occasion.
"I have belief in her, she likes the big stage," Felgate told Reuters. "She enjoys playing, working hard and listens. Over the last year and a half her rise has been amazing.
"A win is great but the performance, that's what you look at. For someone's first main draw appearance at a Grand Slam, the performance was incredible."
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