Facts:
Joe did not get a bonus.
Joe curiously took some half-days off.
Your assumptions:
Joe is planning to leave.
Joe leaving would be a desaster for the company.
If Joe had received the bonus, he would not want to leave.
If the decision was reversed and Joe received the bonus now, he would not want to leave.
Management (George) knows and agrees that Joe leaving would be a desaster for the company.
If it was pointed out to George that Joe wants to leave because of the bonus, he would possibly change his decision.
Your question is, if you should talk to George.
That implies three other questions:
A) What would be best for you.
B) What would be best for Joe.
C) What would be best for the company.
Let's examine your assumptions.
As others have pointed out, this is just your speculation, there are hundreds of possible reasons for Joe's behaviour.
Can you really judge that objectively? It sounds like the company is quite big, so employees leaving and hiring qualified replacement should be rather routine.
Again, that is just your speculation. Even if Joe wants to leave, the bonus may have been just the straw that broke the camel's back, or he simply found a job with more pay, or he wants to move for family reasons, or hundreds other possible explanations.
Dubious. If Joe has already commited himself to new employers or his family, it might be too late. Even if that is not the case, he might be suspicious of the reasons for the change and continue to mistrust the management. And if he somehow learns that you were involved in changing the decision, he might be highly embarassed on how to deal with you in future and decide he prefers not to work close to you anymore.
If George already knew that, why did he not give Joe the bonus?
If George does not consider Joe leaving such a tragedy as you do, it is highly unlikely you can convince him - he probably assumes that as a manager he knows a bit more about such things than an employee. Could you point out any objective facts he is not yet aware of? And even if his private reaction is "my god, what have I done", he might want to hide it and stick to his decision to save face. Managers probably do not enjoy employees telling them how to do their business.
To summarise, talking to George would be effective only if assumptions 1-6 are all true, and some of them are quite unlikely. However, you seem to think "no harm in trying". As the previous answers have pointed out in quite some detail, that is highly naive. That you have good intentions does in no way guarantee a good outcome. You will look like a meddler, possibly like a bumbling fool, possibly like a devious schemer.
Even if you are so selfless to ignore the negative consequences for yourself (A), you might not even be helping Joe (B), because quite possibly he might find another job with better pay or other advantages.
So would you at least help the compay (C)? Not even that is certain - talking to managers about co-workers behind the co-workers' back, even if done in good intentions, will cause mistrust and suspicion between management and employees, and among employees.
Remember, you are not a knight in shining armour riding in to save the day. You are dealing with competent adults who are taking care of their own lifes and their own jobs. In a TV-series meddling in matters which are none of your business tends to have amusing and heart-warming consequences, but this is real life.