"A good memory" vs. "good memories"

I don't have a good memory of my childhood is possible, but I think it's quite an unlikely sentence for a native speaker to produce. If the speaker has an imperfect recollection, then it might be something like I don’t remember my childhood very well. On the other hand, I have good memories of my childhood would, as you suggest, mean that the childhood was a happy one, just as I have bad memories of my childhood would mean that it was the opposite.

There are several different meanings for the word memory, which can lead to ambiguity. Two of these are

the faculty by which the mind stores and remembers information: I’ve a great memory for faces
the mind regarded as a store of things remembered:
he searched his memory frantically for an answer

something remembered from the past:
one of my earliest memories is of sitting on his knee

When you say good memory, it is not clear as to whether you are referring to the effective quality of your recall faculty (meaning 1) or the affective quality of what is being recalled (meaning 2).

When you use the plural form, memories, the second meaning is clear since the first meaning is almost always singular.

If you wish to convey a high level of recall, I would suggest one of the following alternatives: