No. You should take at 'face value' the ANOVA result that the factor does not contain significant differences. Although not common, it could happen that an ad hoc "test" would "find" a bogus significant difference between two levels of a non-significant factor.
Depending on circumstances, this might be due to using a different measure of differences than in the main ANOVA or it might be a chance 'false discovery' due to multiple testing of the same data.
Properly done, you should not even look for such differences, but if you do look and think you have found something, you should ignore your mistake.