Yes. Consider the following (somewhat contrived) scenario.
There are 300 voters and 5 candidates. The election uses the formula (number of votes) / (seats to fill) to determine the quota. Votes are as follows:
AD - 100
BE - 100
CE - 25
CD - 15
DC - 35
EC - 25
(Where "XY - Z" means Z people voted X as their first choice and Y as their second choice)
With 3 seats to fill, the quota is 100. A and B both reach that with no excess votes at all. This leaves C, D, and E with 40, 35, and 25 first-choice votes respectively.
Next, the candidate with the smallest number of votes (E) is eliminated and their votes are transferred to their second choice C. This process continues and D is eliminated transferring all their votes to C. The result is that C now has 100 votes and meets the quota.
Now let's try again with 4 seats to fill. This time the quota is 75, A and B still hit this, but they now each have 25 excess votes which are transferred to D and E. Now we have C, D, and E with 40, 60, and 50 first-choice votes respectively. C is now the candidate with the fewest votes so they are eliminated, transferring their votes to D and E who are elected.
There are other formulae that can be used to calculate the quota, but similar situations can be set up for them.