I am using the GLM Summary in R to determine the significance of the variables in the logistic regression model. I am trying to figure out if there is a way to flip the null and alternative hypotheses such that the null is that a certain variable is not insignificant in the logistic regression model and the alternative is that a certain variable is insignificant in the logistic regression model. This way, if I reject the null, I can accept the alternative with some level of significance. Is there a way to do this? Or is there some other test or model I can use?
No, this is not possible. To show if a parameter $\theta$ (in your case a population effect on the log-odds) is exactly 0 would need an infinite sample size.
You can perform a so-called equivalence test though. It works as follows: Before the analysis, you specify a range $R$ for the $\theta$ which you would judge as being "similar to 0", e.g. $R = [-0.1, 0.1]$ (depending on the meaning and scaling of the variable in question). Then you let the software compute a 90% confidence interval for $\theta$. If it is fully contained in $R$, then you can claim "similarity to 0" at a level of 95%. (It is somewhat unintuitive why a 90% c.i. is sufficient, but that's life.)
Depending on your research question, this can be quite useful.
Comment: Your sentence the null is that a certain variable is not insignificant is not very good. The word "significance" (in the statistical sense) can not be part of the formal definition of the hypothesis. It is an attribute of the data, not of the parameters.