You rarely, if ever, should speak to their manager (alone).
99%+ of the time the issue can be resolved between yourselves and you shouldn't need to get management involved at all. You can do so by:
- Better justifying your reasoning.
- Asking questions to get them to better justify their reasoning.
- Compromising. This is probably the most important one. Try to find some middle ground if possible. Also, many issues aren't all that important and for those it's usually best to just let the other person get their way if you can't convince them. This could go a long way to helping you get your way for some particularly important future issue.
- (Politely) making it clear that you won't compromise.
If that didn't work and you really need to get management involved, the way to think of it is getting them to help the two of you resolve your issue, not trying to get your coworker to do what you want.
If you go to their manager (alone), that very much creates the impression that your goal is to get your way by getting your coworker to do what you want rather than to find some solution that's acceptable to both of you.
In general you should speak to management together, as this allows your coworker to also share their reasoning rather than having you just share your side of the issue. You also don't have to go speak to their manager specifically (it could also be your manager), as "help us resolve our issue" isn't specific to either of you, so on the surface there's little reason to favour one manager over the other. I would probably propose the option of either manager to your coworker rather than just one of the two.
You can speak to your own manager if your coworker refuses to go with you to speak to management, or really at any point before then. Your manager is there to help you resolve issues you encounter in the workplace. This may involve resolving the issues for you, or simply giving advice in terms of what you should do to resolve them yourself. If your manager isn't any help, you can consider asking a different trusted manager for advice, or asking how to resolve your specific situation on this site or a similar one.
If you've tried all of the above and somehow you still haven't found a resolution, then perhaps you can consider speaking to their manager. Since you should already have brought up speaking to management at this point, it would also make sense to tell them you'll speak to their manager. But do make it clear that the goal is to help you find a way to resolve the issue. This may also help change their mind about going with you.
The above applies mostly to conflicts involving differing opinions (broadly speaking).
If you have a "conflict" where they say they're unable to help you with something because they don't have access to some system, for example, this is more an issue of finding the right person to help you (or getting your coworker the access they need) rather than resolving an issue between the two of you. If you believe their manager is the right person to talk to, then do so. Telling your coworker who you're going to talk to, or asking them who you should talk to, would make sense.
If the nature of the conflict is them creating a hostile work environment (e.g. they are harassing you), then that's one of the few times where going to HR would make sense.