As someone who has worked to learn this for years - good luck

5 quarters really is not enough data points for time series. The rule of thumb is you need at least 50 points givens issues such as seasonality. If you can run this as days or months that would be much better.

The first thing to test with time series is if you have stationarity. Dickey Fuller is the most common test [all of the stationarity test have power problems so it is often recommended you run a test that the null is stationarity and the null is not stationarity]. I am not sure with so few data points it is even reasonable to test for stationarity.

I have not seen time series used as you are suggesting. That said I would do a Durbin Watson, or the more advanced test for serial correlation. If there is none do a normal statistical test. If there is serial correlation than you have to use an approach such as Regression with correlated error. None of this works with non-stationarity, you have to transform [difference] the data first.