The 12V relay coil is controlled by a 12V controller but the 12V relay's contact is connected a 24V power source (so will it cause any issues?).

 

THAT depends on the rating(s) of the contacts. You do have that data – right?

 

it just says a 12V relay. It hasn't specified rated voltage for the relay contacts. The relay part number is B120-7022

 

it just says a 12V relay. It hasn't specified rated voltage for the relay contacts. The relay part number is B120-7022

Who is the manufacturer? Never mind.
Part is obsolete and no datasheet is available from Eaton. So I guess you're SOL.

 

My read is it's an obsolete part made by Bussman Eaton. I can't find a data sheet for it.

Ron

 

One site describes it as a 12V, 20 amp SSR.
Genie #1279522GT

 

Mostly a relay coil voltage and contact voltage are quite separate. The only exception I am aware of is older automotive horn relays that only have three terminals. So a relay intended for 12 volt application will work quite well switching 24 volt power. Coil and contact ratings are quite separate.