Do colleges in Mumbai such asSPIT, KJ Somaiya and DJ Sanghavi have management quota?

All the 3 colleges you mentioned admit students through the central admission process (CAP) rounds which are undertaken by the Directorate of Technical Education, Maharashtra (DTE). 

The admission process through cap rounds is always based on merit. If there are seats vacant after the 3 cap rounds, then colleges may admit the students through the management quota.

Management quota is not applicable before the end of round 3. 

The major drawback is that by the end of round 3, mostly all the seats of the reputed branches are taken. So I won’t recommend you wait till the end of 3rd round unless you are planning to pursue chemical or production engineering.