No, the regular future tense does not indicate willingness.
In Latin, the most basic verb for this purpose is velle together with its cousins nolle and malle. For example, "We don't know what to do about Paul, he won't take his medicine" = De Paulo quid agamus nescimus, medicinam accipere non vult. (If you said: ... non accipiet, you would be making a prediction that would by itself say nothing about Paul's willingness.)
There are other possibilities; for example "I am ready, prepared to do something" could be expressed as: paratus sum + infinitive.
Having said that, there is also the periphrastic future, which on first glance is only a grammatical quirk that lets you work around the problem that there are no future infinitives and subjunctive forms, but on second glance can carry a variety of nuances, including: "... denoting an intention to do something. This intention may arise either from the person's own will , or from outward circumstances ..."
Georges gives the following example:
Caesarine provinciam tradituri fuistis an contra Caesarem retenturi?
... which is probably best translated as:
Were you going to hand the province over to Caesar, or were you going to keep it against Caesar?
(Okay, that's no real future tense, but never mind that.)
Of course that's just a nuance, and it is often unavoidable that talking about the future can be read as talking about willingness; if someone says "Numquam in patriam redibo," then it may be wistful or defiant. Depending on context, it may say something about willingness or not.