Yes, they do provoke an Attack of Opportunity at 10 feet
Let's look at the exact wording of the feat:
While you are wielding a glaive, halberd, pike, or quarterstaff, other creatures provoke an opportunity attack from you when they enter your reach.
This establishes a condition and an effect if that condition is met. The condition is:
While you are wielding a glaive, halberd, pike, or quarterstaff [...]
That condition is met if you are wielding a quarterstaff and a whip.
The effect is:
[...] other creatures provoke an opportunity attack from you when they enter your reach.
Since the condition is met, you receive that effect, which means that other creatures provoke Opportunity Attacks when they enter your reach. Your reach with a whip is 10 feet, so creatures provoke an OA when they enter that reach.
If WotC didn't want the feat to function in this faction, they would have changed the wording on either the condition or the effect. I've provided two examples of how that might work:
While you are wielding only a glaive, halberd, pike, or quarterstaff, other creatures provoke an opportunity attack from you when they enter your reach.
This wording would change the condition so that it's no longer met if you're holding any other weapons - it would require you to hold only an eligible polearm to receive the effect.
While you are wielding a glaive, halberd, pike, or quarterstaff, other creatures provoke an opportunity attack from you when they enter your reach with that weapon.
This wording would change the mechanics of the effect so that enemies only provoke attacks of opportunity when they enter the reach of the polearm, rather than entering your reach with any weapon.
The writers made a deliberate choice not to restrict the condition (require wielding only a polearm) or the effect (requiring the reach or attack to be with a polearm).
For a similar example, see Crossbow Expert. The feat includes a benefit to ranged spell attacks. By looking at the name of the feat, you might reasonably assume that benefit was only meant to apply to crossbows. However, the writers explicitly chose not to include wording restricting the effect to crossbow attacks, which allows a feat named "Crossbow Expert" to benefit spellcasting. This was confirmed in the Sage Advice Compendium:
Is it intentional that the second benefit of Crossbow Expert helps ranged spell attacks? Yes, it’s intentional. [...] When designing a feat with a narrow use, we consider adding at least one element that can benefit a character more broadly [...]
They intended for the effect to extend to spellcasting, but they didn't explicitly spell it out. Rather, they explicitly avoided restricting the benefit to crossbows only. If they had intended for it to only apply to crossbow attacks, they would have said so.
If it's reasonable for a feat entitled "Crossbow Expert" to benefit ranged spell attacks, it's reasonable for a feat entitled "Polearm Master" to benefit whips.