As a general rule, these verbs followed by an assertion/proposition/verbal phrase can always be followed by 'that', and native speakers omit 'that' because it is easier/quicker to say. It is very common to omit 'that' with frequent verbs (know, say, think, etc.) and less common with more formal/ longer words (inform, convince). If in doubt, I would keep 'that' because that will always be correct.