No, this is not possible, as the vectors used for the vaccination cannot replicate anymore. Some of the genes necessary for this step have been removed from the viral genome to prevent the uncontrolled spread and replaced by the spike protein which is used as the antigen for the vaccination.
To replicate the virus in the production process, a special cell line with a plasmid carrying the genes necessary for the viral replication has to be used, afterwards the virus is purified and outside of these special cells not able to replicate.
For an overview about the adenovirus type 26 based vaccines, see here:
* Vaccines based on replication incompetent Ad26 viral vectors: Standardized template with key considerations for a risk/benefit assessment