Does it matter which database you connect to when querying across multiple databases?

I'm building a Sharepoint application with Nintex workflow that runs a single SQL query over multiple databases (on the same MS SQL server). Does it matter which database I specify in the connection string in terms of speed? So if my query looks like this:

col1, col2 FROM db1.table UNION ALL

col1, col2 FROM db2.table UNION ALL

col1, col2 FROM db3.table

would it make any difference if my connection string looks like this:

=***;Database=db1; Integrated Security=SSPI; Connection Timeout=900

or this

=***;Database=db2; Integrated Security=SSPI; Connection Timeout=900

or this?

=***;Database=db3; Integrated Security=SSPI; Connection Timeout=900

The table from db1 has more records than db2, which has more records than db3. EDIT: My query is actually more complex than what I wrote above, I just simplified it because I didn't know that would matter. The real queries have a

clause and a

JOIN

on a fourth database (db4). The compatibility level of db2 and db3 are

Server 2008 (100)

, for db1 and db4 it's

Server 2017 (140)

.