It's not "wrong," but it's probably a bad idea. One problem is that you don't have lags for your first ten observations, so you can't use those in your analysis, effectively making your data set smaller.
There are certain lags that we think make sense intuitively: Last period probably effects this period, this time this year is probably related to this time next year due to some seasonal variation patterns. One lag and four lags for you would make sense. Having two years out influence what happens today would be surprising and 2.5 years (or 10 quarters) seems stranger still.
I would chalk up a significant lag at quarter 10 to chance, rather than a good model. Including this lag can lead to overfitting. If you overfit, you will have trouble with out-of-sample forecasting. As a test on this, you might run the model with the 10 quarter lag on the first and second halves of your data to see if you still get a significant/similar result.
Lastly, except in the cases of intuition, I don't like to include one lag, then skip a bunch, then include another. For example, including lags 1 and 4 makes sense intuitively, so that's fine, but adding lags 1, 4, and 10 just seems strange.
Time series is as much art as science, so it does take some playing around.