Yes you can get married at the Philippine Embassy in Qatar provided that youre both single and have proof of it, so you need to get a CENOMAR of Certificate of No Marriage from the NSO in the Philippines (both of you) and make it authenticated (red ribbon) at DFA in the Philippines. Since both of you are here in Qatar a relative can get the said CENOMAR on your behalf provided that you'll give them authorization and have valid id's.

You also need an NOC from your employer and authenticated (red ribbon) birth certificate from NSO in the Philippine, same with CENOMAR a relative can get this for both of you.