Who should I contact if I'm having a conflict with my colleague?

Your Line Manager. ( Before that you should solve it within yourself and colleagues with broad mind discussion )

I would first try to approach my Colleague and try to solve the problem professionally. If that doesn’t work out well then I’ll talk to an assistant manager or the manager. 

I would try to resolve the conflict with my colleague in first place

1 My Manager 

2 My Head 

3 My colleagues

1- your colleagues 

2- the head 

3- your manager 

your immediate boss, HR

1. The colleague you are facing issues with.
2. If that still does not solve, another member within the team who would be fair and just on both sides.
3. Next resort would be talking to your supervisor/manager and try keeping that as the final resort.

Avoid escalating the issue further as it would not create a good impression as a team.

Any work and business problems has to be solved within the company , directly with the colleague or someone who is competent to solve problems at work, for example someone from HR department.

Any private problems, disagreements or  disputes should not have an impact on work or working environment and this kind of problems should be solved privately in a nice way.

with your selfe, in work no pleam culture 

Better to solve mutually, with a professional behaviour.

1. Your colleague (Try to resolve the issue professionally)
2. Line manager/supervisor
3. HR department
4. Boss (must be the last options when everything fails)