How do people in Real Life fall in love? They get to know each other. They share some interests, so they enjoy spending time together, and have common things to talk about. They respect each other. Each has qualities the other finds positive, perhaps even admirable. They are willing to forgive each other's less pleasant attributes.
To make a fictional romance believable, you would have to show those elements: they would have to have positive interactions with each other. Each would have to earn the other's respect.
A relationship with no arguments, no negative interactions, is impossible. You would have to show how those interactions ultimately lead to constructive rather than destructive results - better understanding.
We expect two partners to bring in equal measures to the table - otherwise the reader might well ask why would character A wish to remain with character B. So consider what qualities each character possesses that the other does not, what each contributes to the relationship, in what ways they are together stronger than each one on their own.
You can show the process of falling in love by way of the characters each thinking of the other while they're not there, being aware of each other's presence, contemplating what the other's opinion of them would be on a given situation, etc.
Consider, for example, Pride and Prejudice:
No sooner had [Mr. Darcy] made it clear to himself and his friends that she had hardly a good feature in her face, than he began to find it was rendered uncommonly intelligent by the beautiful expression of her dark eyes [...] and in spite of his asserting that her manners were not those of the fashionable world, he was caught by their easy playfulness. [...] He began to wish to know more of her, and as a step towards conversing with her himself, attended to her conversation with others. (Jane Austen, Pride and Prejudice, chapter 6)
That is the very beginning - Darcy notices Elizabeth - that is, notices her as standing out from the crowd. The two proceed to interact with each other, they are more and more on each other's thoughts, they grow to understand each other better. Through actions, they change, and change each other's opinion of themselves, and grow to accept each other's weaknesses.
You also need to show a couple's understanding each other - how they fit. Not completing each other's sentences - that trope has been overused, but taking into account each other's views without being reminded, understanding each other's intent. An example of a couple understanding each other, again from Pride and Prejudice:
'What made you so shy of me, when you first called, and afterwards dined here? Why, especially, when you called, did you look as if you did not care about me?'
'Because you were grave ans silent, and gave me no encouragement.'
'But I was embarrassed.'
'And so was I.'
'You might have talked to me more when you came to dinner.'
'A man who had felt less, might.'
'How unlucky that you should have a reasonable answer to give, and that I should be so reasonable as to admit it!' (ibid, chapter 60)