Can I claim a dependent if they file their own tax return? They are over 19 and not a student but they don't have any income

How did they file a return if they didn't have any income?  Is this your child?  You can claim someone if they lived with you the whole year and made less than 4,300.

See IRS Publication 501 starting on page 11 bottom Dependents

https://www.irs.gov/pub/irs-pdf/p501.pdf

Who can I claim as a dependent?

https://ttlc.intuit.com/community/credits-and-deductions/help/who-can-i-claim-as-my-dependent/00/26781

No, you cannot claim a person as a dependent if they filed their own return and claimed himself (they did not check the box on form 1040 that says someone else can claim them as a dependent).

But, that person can file an amended return, to un-claim himself to allow you to do so.  But, he may have to pay back any refund he received.  Things are particularly complicated theses days with the stimulus payments.

For more specific assistance, you really need to describe your situation. 

There are two types of dependents, "Qualifying Children"(QC) and Other ("Qualifying Relative" in IRS parlance even though they don't have to actually be related). There is no income limit for a QC but there is an age limit and student status test, a relationship test and a residence test. Only a QC qualifies a taxpayer for the Earned Income Credit and the Child Tax Credit. The Other dependent (qualifying relative) credit is worth (up to) $500 per dependent and is non-refundable.  That is, it can only be used to reduce an actual tax liability.

A person can still be a Qualifying relative dependent, if not a Qualifying Child, if he meets the 6 tests for claiming a dependent:

1. Closely Related OR live with the taxpayer ALL year
2. His/her gross taxable income for the year must be less than $4300 (2021).
3. The taxpayer must have provided more than 1/2 his support

In either case:

4. He must be a US citizen or resident of the US, Canada or Mexico
5. He must not file a joint return with his spouse or be claiming a dependent of his own
6. He must not be the qualifying child of another taxpayer

See full dependent rules at: https://turbotax.intuit.com/tax-tools/tax-tips/Family/Rules-for-Claiming-a-Dependent-on-Your-Tax-Return/INF12139.html