Can I use quotation marks to suggest that somebody should say something?

Yes, I believe that it is correct grammatically, and intuitive as well. It is the only way to advise about using a particular phrase or expression. For example, you could have said:

You should greet your neighbors.

That sentence has a different meaning. The only way to convey specifically what you meant is to say:

You should say "Hello" to your neighbors.

or "Good morning" or "How do you do?" or "Good day" or whatever salutation you feel is best.

EDIT
I realize that this is not Etiquette Stack Exchange, but Emily Post is known for her etiquette and correct grammatical usage. The online version of her work has an entry on this subject which is consistent, with "Hello" in quotation marks.

You might not want to capitalize the "h" in "hello", but I am less certain about that. Personally, it looks right to me (your question was not about that anyway, but rather, the use of quotation marks).

Of course, you should.

You are 'quoting' them in a way pre-facto.

Citing an article:
But speaking in Birmingham, Mr Cameron will tell the Pope: “Faith is part of the fabric of our country. It always has been and it always will be.”

(a) You specify the use of double inverted commas. I assume that you are asking whether the convention is that what goes inside them has to be 100% accurate or the sky will fall on our head, and with Kris' example, for instance:

'But speaking in Birmingham, Mr Cameron will tell the Pope: “Faith is part of the fabric of our country. It always has been and it always will be.” '

we are risking our future on Mr Cameron getting his lines right. Perhaps a DV would be appropriate.

In fact, though I try to conform to the rule 'keep words within double inverted commas as verbatim copies of the source material', with fictional writing and recordings of future conversations etc, context indicates that there is no absolute guarantee of (or, for fiction, real meaning for) 'verbatimity' (the word should exist). One cannot guarantee that Mr Cameron will even get to meet the Pope (though it may now already actually have happened!). To be sure the sky will stay in place, one could use a hedging device such as: 'According to sources near to Parliament [ie the man sleeping on the bench outside], Mr Cameron intends to tell the Pope when speaking in Birmingham: “Faith is part of the fabric of our country. It always has been and it always will be.” '

(b) I've addressed before - though it may have been on a different website - the general relaxing of rules governing the requirements of quote structures. Punctuation guidelines (single or double inverted commas? introductory comma, colon, zero punctuation mark? siting / doubling of terminal punctuation?) vary widely, both inter- and intranationally.

With regard to quote verbs (those used to introduce direct speech - and any other verbatim record, as for instance a quote from a poem, book or notice), many more are allowed nowadays, including a zero verb-and-ascription where confusion would not arise:

"Hello!" said John.

"Hi!" smiled Sally.

John smiled back. "Are you going to the Christmas market?"

"Yes - are you?"

Report and quote structures that are identically worded are often used also:

You should say hello to your neighbours.

You should say "Hello" to your neighbours.

The first one would be more normal, a general recommendation not to be stand-offish, with no recommendation as to which exact manner of greeting should be chosen. As Feral Oink says, the two versions are not synonymous, the second advising on the exact wording recommended. Convention decrees that the start of the quote in this version be capitalised (the start of an [intended] sentence equivalent); it would be unnecessary - perhaps wrong - to capitalise in the non-quote version. Say hello to is even given idiom status at TheFreeDictionary - a transitive multi-word verb with the near-synonym greet. Say goodbye to is a more opaque idiom, as in Say goodbye to those blocked drains! We wouldn't use a quote structure here, though early adverts may have done.