Let us assume that the PCs agree to play a game in which there is a quest. They must, I think, therefore be in favour of the general aims of the quest. Does it necessarily follow, however, that all of them must be in agreement over the methods used to fulfil the quest?
A thought experiment, once more based on Lord of the Rings (not the first time I have done this I admit). The quest in Lord of the Rings is to destroy the Ring by taking it to Orodruin. However, Boromir, in 'Minority Report' style, departs from the consensus and wants to use the Ring as a weapon against Mordor. Both viewpoints are about fighting Sauron, one by destroying a powerful artefact that could be used by the baddies, the other by using the artefact against the baddies. In this thought-experiment version, Boromir takes the Ring from Frodo at Amon Hen. What happens next?
Boromir has fallen under the baleful influence of the Ring, but hasn't changed sides; he is still opposed to Sauron, and wants to use it as a weapon against Sauron for the good of Gondor (as he sees it). He's not going to ally himself with the Orcs in the woods, they and he are still still enemies. The rest of the party... err, Fellowship... wants to 'rescue', not fight, him, even if they fundamentally disagree with his perspective. So, Boromir is still fighting the Orcs, but not allowing the Fellowship to stop him taking the Ring to Minas Tirith. He battles his way to Gondor, and from there the plot takes a different turn as the now Ring-wielding Gondorian forces both resist Sauron and are corrupted from the inside.
There are some successes for Gondor due to wielding the Ring but also increasing jealousy between Boromir and Denethor, as well as paranoia in the Gondor command about 'internal enemies' (not helped by Faramir throwing in his lot with Aragorn's 'loyal opposition'). This leads to a nightmare 3-way war between the forces of Sauron, the Ring-wielding Gondorian faction around Boromir (after a power-struggle, Boromir kills Denethor and takes absolute control) and the Aragorn/Faramir faction, waging guerrilla war from Ithilien, fighting off the Gondorian loyalists as well as Mordor, but also trying to unite all the anti-Mordor forces. Ultimately, Aragorn's guerrillas break into Minas Tirith and steal the Ring before racing to destroy it in Mt Doom. Boromir, perhaps, is unsavable, and resists the Aragornian 'rebellion' to the end, but ultimately the Ring is destroyed and Gondor saved.
Is something like this a reasonable version of 'the Quest'? It's not Tolkien's story but is it a reasonable, workable, even playable departure from it? I think it is. It's maybe a bit more 'Game of Thrones' (or even 'Star Wars') than 'Lord of the Rings', but I can see how it might work. There's no reason why this couldn't be the unfolding story of a game based on the starting-point of 'The Lord of the Rings'.
In the Quest for the Relics of McGuffin, the Sages of the Unpr'Onounc'Eable Temple want the Relics brought back to the Temple so they can use them to defeat Lord Doombad. Let's say the NPC Sage Andonion has roped in Scrofula the Thief, who is a PC, to bring back one of McGuffin's relics, the Spoon of Density, from the Dank Citadel. However, Scrofula, having gone to to the Dank Citadel with his mates (the rest of the party) and liberated the the Spoon of Density, decides he wants to go off-piste and uses it to attack Lord Doombad instead of taking it back to the Sages. Is this reasonable?
It might not be in a novel (I don't know, actually maybe it would) but it should be in a game. A quest does not necessarily imply unanimity about methods, just aims. In 'Lord of the Rings' everyone wants Sauron defeated, but they disagree about how that is best done. That should be OK in a game too, and it should be possible to change course to fulfil the quest. The Sage Andonion has engaged the PCs - somehow! (and that 'how?'still requires some thinking about) - in a quest (getting the Spoon from the Dank Citadel will somehow help defeat Lord Doombad... either because the Spoon is a powerful artefact and it will help to add its power to the good side, or to deny its power to the evil side) but the PCs have autonomy to carry out the quest in the way they see fit. How they see that will depend on the setup, how the information is actually given to them, and how much they trust the veracity or sagacity of their quest patron.
But probably I need to keep picking away at those things, so I'll leave this here for the moment.
Happy Christmas, and whatever else you may be celebrating.