Yes, and no.
"Telefonálni" is used without a direct object. It is not even used with the definite conjugation (usually).
You can say "valakiVEL telefonálok" - with someone. Or "telefonon beszélek valakivel" - I am speaking on the phone with someone.

And you can say "telefonálnom kell" - I need to make a phone call.

But if you want to call someone, you say "felhívni valakit" - literally, "to call up" somebody.

"Holnap felhívlak." - I will call you tomorrow.
"Felhívhatlak?" - Can/May I call you?

This is a good example where dropping the preverb carries the present continuous sense:

"Kit hívsz?" - Who(m) are you calling? (on seeing someone making a call)
"Hívom a rendőrséget" - I am calling the police.

To make it complete:
To dial is "tárcsázni"
To hang up is "letenni (a telefont)". Of course, this is with old technology. :)