Being enhanced by the Quicken Spell Metamagic does not change the specific actions that must be taken in order to Counterspell.
First, let's look at the steps required to perform a Counterspell
You must select an opponent as the target of the counterspell, which is done by choosing to ready an action (this is a standard action)
If the target you selected tries to cast a spell, you make a Spellcraft check (a free action). If you exceed the DC, you identify the spell and can proceed with your counterspell attempt, otherwise, you can't.
You decide if the spell that was just cast is a spell you want to counterspell, then may attempt to counterspell it.
We also know (from the section that you quoted) that Metamagic doesn't affect whether or how a spell can be countered, or whether a spell can be used to counterspell.
So what affect would Quicken Spell have when used in conjunction with Dispel Magic?
Well, it wouldn't actually do anything. Quicken Spell changes Dispel Magic's casting time from 1 Standard Action to 1 Swift action. But we aren't casting Dispel Magic on our turn, so the type of action used to cast it is effectively irrelevant. Our Standard Action is being used to take the 'Ready an Action' action, not to cast Dispel Magic. Our Swift Action doesn't come in to play in this scenario either, since we don't "pre-cast" Dispel Magic and we can't choose to Ready as a Swift Action, it must use our Standard Action. To be more specific, we use a Standard Action to Ready an Action, but the type of action we ready can be Standard, Move, Swift, or Free. In this scenario, it would be pointless to ready a Swift Action to cast Dispel Magic with the Quicken Rod, since we don't gain any additional benefit from casting as a Swift instead of a Standard, it would just be a waste of a charge from the rod.
OK, so what does that segment actually mean then?
Well, it's pretty straightforward: having a metamagic affect applied to a spell doesn't change whether it can be counterspelled or used as a counterspell. For example, let's say Wally the Wizard has Empowered Fireball prepared. He's facing Sally the Sorcerer in a duel, and he decides that he's going to attempt to counterspell Sally's next spell. Sally casts Fireball. Assuming that he passes his Spellcraft check (good odds, since he's a wizards), since Wally has Fireball prepared, he can immediately attempt to counter Sally's Fireball spell, even though his Fireball was prepared with the Empower Spell Metamagic. This works in the reverse as well, should Sally decide that she wants to attempt to Counterspell Wally, and he decides to cast Empowered Fireball, she can still counterspell him even though her Fireball isn't Empowered.
Dispel Magic works the exact same way as any other spell used as a counterspell, with the only difference being the added step of the Caster Level check used to determine if you successfully counterspell, rather than the automatic success from using the exact same spell.