Neural network q learning for tic tac toe - how to use the threshold
You are effectively implementing $\epsilon$-greedy action selection. 

The usual way to represent this in RL, at least that I am familiar with, is not as a "threshold" for probability of choosing the best estimated action, but as a small probability, $\epsilon$, of not choosing the best estimated action. 

For consistency with RL literature that I know, I will use the $\epsilon$-greedy form, so instead of considering what happens as your threshold rises from 0 to 1, I will consider what happens when $\epsilon$ drops from 1 to 0. It is the same thing. I hope you can either adjust to using $\epsilon$ or mentally convert the rest of this answer so it is about your threshold . . .

When monitoring Q-Learning, you have to be careful how you measure success. Monitoring the behaviour on the learning games will give you slightly off feedback. The agent will make exploratory moves (with probability $\epsilon$), and the results from a learning game might involve the agent losing even though it already has a policy good enough to not lose from the position where it started exploring. If you want to measure how well the agent has learned the game, you have to stop the training stage and play some games with $\epsilon$ set to $0$. I suspect this could be one problem - that you are measuring results from behaviour during training (note this would work with SARSA)

In addition, choosing values that are too high or low for your problem will reduce the speed of learning. High values interfere with Q-learning because it has to reject some of data from exploratory moves, and the agent will rarely see a full game played using its preferred policy. Low values stifle learning because the agent does not explore different options enough, just repeating the same game play when there might be better moves that it has not tried. For Tic Tac Toe and Q-learning I would suggest picking a value of $\epsilon$ between $0.01$ and $0.2$

In fact, with Q-learning there is no need to change the value of $\epsilon$. You should be able to pick a value, say $0.1$, and stick with it. The agent will still learn an optimal policy, because Q-learning is an off-policy algorithm.