The situation you describe is pretty complex. I'll pull it apart and look at the different aspects, but the short version is: There should be a way for both of you to get what you want.
To begin with, a player is entitled to control of their own character. Permanent changes to the character, particularly to the character's personality and core concept should be done collaboratively with the player who controls that character.
Think of it this way: if you sat down at the table with a Barbarian you were excited to play, it would probably irk you if two weeks in another player said "you're playing a wizard now." Your character's class was something that was your choice, and that choice was taken away from you. Most people don't like that.
This is not to say that change can't come from sources outside of the player. It routinely does in many campaigns. But if the player is actively against the change, if they feel that change invalidates the character they've created, then that tells you that you are crossing a line.
Alignment, particularly in latter editions of D&D, is not intended to be a be-all end-all definition of the actions a character is allowed to take. Alignment is a rough descriptive guideline for your character, much as your background is.
All that "Lawful Good" really says about a character is that they generally like and support other people, and that they generally like order and consistency. That's it.
So unless this cult is a huge departure from the D&D norm (committing human sacrifices or kicking puppies on a regular basis or similar), it is highly likely that you can justify some reason for a Lawful Good character to be a member.
Making the Tough Decisions
Making the Tough Decisions is an essay by Rich Burlew. I recommend it as reading material for all role players, of all stripes.
The short version of it is that character decisions are ultimately controlled not by any concept of "realism," but by the choices of their players. It is the responsibility of all players to avoid situations that create a deadlock.
You shouldn't insist on changes to his character. If he's not interested in changing, just go along with it.
But at the same time, he shouldn't stand in the way of the story or the group. If the group is joining the cult, he should find a way to make that work for his character. If you are starting a cult, he should find a way for his character to be okay with what your character is doing.
The example I often use of this is Shepherd Book, from the Firefly TV series. He has an extremely strict code of conduct and ethics, a code which is dramatically different from the rest of the crew.
And yet, he is able to participate with the crew without turning every incident into an intra-party squabble. He does not apply his ethics to other characters, and he is able to find ways to bend his rules to join in ("The lord never said anything about shooting kneecaps").
At the same time, the crew doesn't push him to be anything other than what he is. No one demands that he either take up arms or be booted off the ship.
Putting it Together
If joining this cult is truly a direction that a majority of the players want to take, collaborate and find a way to make this work for all the players involved.
Why would joining this cult require an alignment shift? Is there truly no way for a Lawful Good character to participate? How far do they need to bend?
Can you come up with a few specific concessions, and get buy-in from the players and the DM involved? Perhaps the player needs to be a little more gritty, or the cult needs to be a little less evil.
Nuke the Site from Orbit; It's the Only Way to be Sure
Finally, if joining this cult is a radical departure from the existing game, it may be better to stick a pin in it. Come back to it later, when the first campaign is done (or as a parallel campaign). Give everyone a chance to build a character that fits the cult's play style, or to recuse themselves from the campaign if it doesn't sound fun to them.
Your Specific Situation
Given that this appears to be about adding some flavor and long-term goals to your characters, here is the easiest way to handle it. You say "I try to convert your character to my cult." He says "My character is not converted." He says "I give you a sermon to convert you from your wicked ways." You say "Yeah, that's not happening." You move on with the game.
It is possible to have characters be in conflict, without having the players be in conflict or making that conflict the focus of your campaign. As long as you never actually intend to cross a major line with your cult, and he never actually goes out and smites it, everyone can coexist.
If him joining the cult becomes necessary, there are plenty of reasons for a Lawful Good character to join such an organization:
He joined it to be with his friends.
He joined it to gain power to advance a good cause.
He joined it to try to keep the darker aspects of it in check from within.
He (the character, not the player) is simply unaware of how dark the cult's end game is.
He isn't a member of the cult. He's your friend, and he helps you.