This (Norwegian vegan body builder!) dude would say 'nei!'
I can't answer your question definitively, as I'm still learning Norwegian myself, but the impression I'm getting about the difference between behøver, må and trenger is this:
• å behøve is used to denote when something would be good, healthy, or beneficial for the subject of the sentence. I think it's also an auxiliary verb, to be followed by another verb in the infinitive form: "Behøver jeg å drikker melk?" = 'Do I need to (is it good for me to) drink milk?'
• å måtte (må) is used when something is required by the rules or by an authority, and is followed by another verb in the infinitive form, but without the preceding å: "Må jeg gå på skolen?" = 'Must I/do I have to go to school?'
• å trenge is used with a noun (i.e., not with other verbs) to show when an object is needed or physically required: "Vi trenger tre egg for å lage kaken" = 'I need three eggs to make the cake.'
If a native or fluent speaker could confirm which parts of this I have got right - or if I'm wrong, and some of the words are completely interchangeable - I would very much appreciate it!
Obliged suggest some external pressure so the meaning of the word is different. It's how an obligation and a need mean different things, so do the verbs. You couldn't say I'm not obliged to eat meat to stay healthy, that doesn't make sense. The word behøver translates exactly as need and that would be it.