"quality" here has NO implied meaning. It means characteristic or feature (特性). For example,
The trade mark of Mercede-Benz is symbolic of luxury.
"of their nations friendship" is a prepositional phrase that is used as an adverb to describe "symbolic".
One thing you need to know is:
Prepositional phrase can be used as an adjective or adverb. If you write "a symbol of their nations friendship", then, "of their nations friendship" become an adjective.
Now, why use "symbolic", but not "symbol"?
You need know when you use noun in this sentence, you make "handshake" equal to "symbol". Like you said:
You are a boy. (you = a boy)
However, handshake is NOT a symbol.
Hence, you use adjective, like
You are handsome.
"handsome" is just a description of "you", but is not equal to you. This is why it use "symbolic", not "symbol"
This kind of use is very common, for example:
It is very kind of you. (kind is an adjective)
He is tired from the hike. (tired is an adjective)
2014-09-09 01:56:02 補充：
I just want to advice you:
Don't bring the grammar to analyze the sentence BEFORE you can understand the context. Once you can understand the context, then you find a grammar rule that is suitable for it. Also, use your logic thinking.