I was wondering if the rod of extending could be used as a weapon and possibly used like a quarterstaff

I don’t see why not, although it might depend on circumstances; I don’t think it would be a “magic weapon”, though.

That's what I was thinking, I'm trying to make a character with a compact weapon

 The pole would be classed as an improvised weapon:

Sometimes characters don't have their weapons and have to attack with whatever is at hand. An improvised weapon includes any object you can wield in one or two hands, such as broken glass, a table leg, a frying pan, a wagon wheel, or a dead goblin.

Often, an improvised weapon is similar to an actual weapon and can be treated as such. For example, a table leg is akin to a club. At the DM's option, a character proficient with a weapon can use a similar object as if it were that weapon and use his or her proficiency bonus.

An object that bears no resemblance to a weapon deals 1d4 damage (the DM assigns a damage type appropriate to the object). If a character uses a ranged weapon to make a melee attack, or throws a melee weapon that does not have the thrown property, it also deals 1d4 damage. An improvised thrown weapon has a normal range of 20 feet and a long range of 60 feet.

It is then up to your DM whether the pole of collapsing  is similar to a quarterstaff, some other weapon or no weapon at all.  As it is 10 ft long when it is extended if I was DM I would probably rule it is similar  to a Halberb for damage and proficiency (also making it heavy) but it would deal bludgeoning damage

That'd be up to your DM.