Is it possible to select two baseball teams such that nobody on Team A was ever teammates with anybody on Team B?

I assume it would be very easy. Begin by stocking Team A with AL rookies and Team B with NL rookies. After that you might need to get selective. Unless I'm missing the point...

He probably meant two actual teams. Otherwise it's very easy to assemble, you just need to build a 25 man roster of guys who have never left their league.

White Sox are shopping Quentin around to open up some payroll, and are reportedly looking for bullpen help. Could be a fit.

Oakland and someone? Almost all of their pitchers are homegrown. The guys you'd need to avoid are Wuertz, Breslow, Kouzmanoff, Crisp, Sweeney, CoJack and DeJesus. That's probably doable.

The 1924 Brooklyn Dodgers and the 1973 Montreal Expos.  I'm an idiot.

Well the Cleveland's might be one of the best bets. Of the year end roster, Luis Valbuena had a season and Choo had 30 at bats (albeit over 2 years) with the M's, and Pronk had 70 at bats with the Rangers 8 years ago. Of the pitchers Westbrook got into 3 games with the Yankees a long time ago, Masterson pitched a bit for Boston, Chris Perez with the Cards, and Joe Smith with the Mets. I think relievers are what would make this tough, and the Cleveland's are almost all home grown.

Shelley Duncan played with the Yankees for 3 years and Jayson Nix payed for the White Sox and Rockies, that might ruin them though.

Guess number two... the Marlins, who have an almost entirely home grown team, don't sign free agents, and traded a bunch of veterans at the deadline.

I'd think it would be impossible during the last twenty years at least.  Of the 290 possible match-ups for the Jays from 2001 to 2010, Vernon Wells alone would have played with somebody on maybe half of those teams.

I would say there's probably no team entirely like that, at least as long as we include minor-league teams, college teams, high-school teams, fall league teams... etc.

2010 Indians and Marlins are pretty close, though
Mike Lamb, Brian Barden and Hector Luna cause some problems even though are no longer on the roster.

A reverse test would be fun - connect the Jays of 2010 (or any past year) to every last major league team in as few steps as possible. Using only teams/teammates they were on pre-current Toronto stint, major leagues only.

IE: Buck connects direct to KC, Overbay to Arizona & Milwaukee, A-Gon to Florida & Boston & Cincinnati, EE gets Cincinnati as well, Lewis to SF, Bautista to Pittsburgh & TB & Baltimore (& KC)

So the starting lineup (BR style) covers 10 ML teams, leaving 19 on the list to connect to via pitchers and bench. That is before checking teammates or minor leagues which each would be an extra degree of separation.

Mike Lamb, Brian Barden and Hector Luna cause some problems even though are no longer on the roster.

Not as many problems as they cause when they are on your roster.

The 2010 Jays are directly connected (ie. a 2010 Jays player played for XX in the past) to all but the following teams:

Houston
Philadelphia
Detroit
Texas
LA Dodgers
San Diego

However, there was at least one 2010 player on 5 of these teams that directly connects to the Jays (Wallace, Halladay, B. Molina, Barajas, Eckstein)

The only team without any direct connections to the Jays in 2010 was the Tigers. Sadly, several 2010 Jays played with 2010 Tigers in the past (Miggy and AGon in Fla for example) so we still don't have an answer to the original question.

Taylor Buchholz pitched in 22 games for the Astros in 2006.

John McDonald was traded to the Detroit Tigers for himself in 2005.

Nick Green played in 5 games with the Dodgers this year before joining the Jays.