If a professor shares their office phone number on their website, can I make a call and ask them about postdoc positions? What should I say to them?
Personally, I wouldn't attempt it. If you interrupt something they consider serious they won't like it much. And, what can they say but to look for published announcements? Do you think they want to listen to your qualifications in real time? It is just to easy to hang up, as you might do for any marketing call.
A short introductory email would be preferable. Say who you are. A sentence about your interest. A couple of sentences about your qualifications. Offer to provide any additional information needed. Ask for a reply.
The email can be handled asynchronously, which is much easier. If it is short then it might be read.
But, I suggest saving phone calls for people you already know and who won't mind being interrupted.
And a long and tedious mail will get put on the back burner, maybe forever. Don't include CVs and such in the initial contact. Wait for a follow up request.
In this day and age, it is almost never appropriate to cold call anybody. This includes (but is certainly not limited to) any potential future employers.
Phone calls are a very intrusive way of trying to contact someone, and best reserved for urgent communication that requires an immediate reply. By choosing to cold call someone, you are effectively asserting that your call is urgent. When it turns out it is not (like enquirying about a potential future post-doc position) this is likely to annoy whoever it is that you are calling.
TL;DR Do not cold call people.