Thanks to Baseball Reference, I learned that the answer is yes. Check out these transactions:
July 6, 1986: Traded Duane Ward to the Toronto Blue Jays for Doyle Alexander.
July 6, 1986: Traded Joe Johnson to the Toronto Blue Jays for Jim Acker.
(Yep, they made two one-for-one swaps of right-handed pitchers on the same day.)
February 2, 1987: Traded Craig McMurtry to the Toronto Blue Jays for Damaso Garcia and Luis Leal.
(If ever there were a trade where both sides ended up with nothing, this was it. All three players were veteran Major Leaguers, though Leal hadn't played in the bigs in over a year. McMurtry got injured in Spring Training, spent the season in the minors rehabbing and never pitched a game for Toronto. Former All-Star Garcia also got injured and only made it into one minor league game that season. His total output for Atlanta was 21 games in 1988, where his OPS+ was a dreadful -3. Leal was returned to Toronto before the season began, and he never made it back to the Majors.)
December 5, 1988: Drafted Geronimo Berroa from the Toronto Blue Jays in the rule 5 draft.
December 5, 1988: Drafted Matt Stark from the Toronto Blue Jays in the rule 5 draft.
March 29, 1989: Purchased Mark Eichhorn from the Toronto Blue Jays.
August 24, 1989: Traded Jim Acker to the Toronto Blue Jays for Francisco Cabrera and Tony Castillo.
(Just three years after trading for Acker he sent him back, and in return he got a player who'd deliver one of the most memorable hits in postseason history. Not a bad deal.)
November 20, 1989: Purchased Alexis Infante from the Toronto Blue Jays.
December 17, 1989: Traded Ricky Trlicek to the Toronto Blue Jays for Kevin Batiste and Ernie Whitt.
Each of these players acquired from Toronto was in the organization while Cox was manager, though some were only in the minors at the time. It's clear though, that Cox' familiarity with the Blue Jays' system and front office shaped his decision-making.
For more on the subject of Cox the GM, I'd recommend this excellent piece. It not only demonstrates how much bigger a role Cox had in building the Braves than most people realize, it challenges the commonly-accepted truism that John Schuerholz was one of the greatest GM's of all time. Did you know that Schuerholz was reviled in Kansas City before he came to Atlanta? I certainly didn't.
Of course, Schuerholz wasn't the only one whose reputation was rehabilitated by the Braves' turnaround in 1991; the other was Cox himself. In 1990 the Braves were still a losing team, and as far as fans were concerned, he wasn't getting the job done. It's funny to think about it today, but when Cox took over as manager many people saw him as a placeholder. In August 1990 Atlanta Journal-Constitution columnist Terence Moore suggested that the Braves hire Joe Morgan as their new manager (the Hall of Fame second baseman, not the guy managing the Red Sox). In response to that suggestion a fan wrote this amusing letter to the editor (which is actually what inspired me to write this piece):
Amazing, isn't it? 20 years later, Cox has a future in the Hall of Fame, the Murphy trade is recognized as the right move (yep, Cox managed Dale Murphy in Atlanta) and fans are still annoyed by Morgan's announcing. Long-term positives can enhance our view of an unpopular situation, but some things never change.Hiring Joe Morgan to run the Braves would be about a [sic] big a mistake as hiring Bobby Cox or trading Dale Murphy. If one were to listen to Morgan announce on ESPN, he or she would realize Morgan's level of knowledge of the game. Just because a person has the God-given talent to play doesn't mean he has the ability to manage. Look at our Henry Aaron.
Chris Small, Atlanta