No. The tests generally use antibodies to test for your antibodies. The only antigen would come from you. And as noted by others, even if the test did contain antigen, it would 1) likely not be infectious on its own and 2) not enter your body.

The type of test you’re thinking of is a lateral flow immunoassay or dipstick test. They perform a type of test (assay in fancy scientist speak) called a sandwich assay. The strip is prepared with three things:

1. a loading pad that includes an antibody which binds to the target analyte (the thing you are testing for). We will call this antibody A. For the type of test you describe, an antibody test for HIV, the analyte is human anti-HIV antibodies. The Antibody-A’s are bound to something that generates a visible color, often a metal particle such as gold.
2. A test line, which is a line we’ve drawn across the strip and chemically glued another set of Antiboy-A’s down so they are stuck to the strip
3. A control line which has a second type of antibody (Antibody-B) which will bind to Antibody-A

When we add a sample to the loading pad, it will start flowing down the strip. If analyte is present it will bind the loose Antibody-A’s and become colored as it flows down the strip. When it hits the test like those Antibody-A’s will bind the analyte too. So if analyte is present we form a “sandwich” with two Antibody-A’s as the bread (one bound to a colored marker and one bound to the strip) and the analyte as our filling. If there’s no analyte (or if there is and there are extra Antibody-A’s) they will get grabbed by the Antibody-B’s at the control line.

PS: Antigen not Antigene. Antigen just means something an antibody binds to. Doesn’t have to be (and usually isn’t) anything genetic.