Does 'to believe that P' imply 'to believe that P is true'?

If we rephrase this question to follow Jo Wehler's excellent suggestion, we get:

Does 'to believe that p' imply 'to believe that "p" is true'?

First, the answer to the re-phrased question is "no", since to believe is to have a belief and one can have a belief that p without having a belief that "p" is true, in particular if one doesn't have a concept of truth, or indeed a concept of proposition.

However, the re-phrased question may not be equivalent to the original question from its author's point of view. The following might be closer to it:

Does 'to believe that p' imply 'to believe that it is true that p'?

First, there is a sort of marked conceptual distance between "It rains" and "It is true that it rains". The latter wording suggests an abstraction on the part of the believer.

Second, although "It is raining" does imply "It is true that it is raining", we can believe the former without even having any thought corresponding to the latter.

The distinction, however, comes with the word "belief", which refers to a psychological state, not to any widely accepted logical concept, unlike for example the word "true".

A valid implication involving the word "belief" would have to include a formal and logically consistent definition of the term. Minimally, you could just posit the implication 'to believe that p' implies 'to believe that it is true that p'. That would help in making valid arguments about beliefs but, then, how would you justify your assumption that this implication is true?

Does 'to believe that P' imply 'to believe that P is true'?

If we rephrase this question to follow Jo Wehler's excellent suggestion, we get:

Does 'to believe that p' imply 'to believe that "p" is true'?

First, the answer to the rephrased question is "no", since to believe is to have a belief and one can have a belief that p without having a belief that "p" is true, in particular if one doesn't have a concept of truth, or indeed a concept of proposition.

However, the rephrased question may not be equivalent to the original question from its author's point of view. 

The following rephrasing might be closer to Bernard Williams' original idea:

Does 'to believe that p' imply 'to believe that it is true that p'?

First, there is a sort of marked conceptual distance between "It rains" and "It is true that it rains". The latter wording suggests an abstraction on the part of the believer.

Second, although "It is raining" does imply "It is true that it is raining", we can believe the former without even having any thought corresponding to the latter.

The distinction, however, comes with the word "belief", which refers to a psychological state, not to any widely accepted logical concept, unlike for example the word "true".

A valid implication involving the word "belief" would have to include a formal and logically consistent definition of the term. Minimally, you could just posit the implication 'to believe that p' implies 'to believe that it is true that p'. That would help in making valid arguments about beliefs but, then, how would you justify your assumption that this implication is true?