Can I complete The Blessings of Nature without hurting the tree?

As part of the quest line The Blessings of Nature I have acquired a specialist weapon by the name of Nettlebane that I am supposed to use to chop through the roots of the Eldergleam tree to obtain some sap.

In typical fashion it turns out there are people inside the tree's sanctuary area, who make various comments about how beautiful and awesome it is, and so on.

This makes me suspect there is some alternative way for me to complete this quest that doesn't require me to hack apart the tree. Is there? And if so, what is it?

There is!

During the quest, you should've run into Maurice Jondrelle, who asks to come with you. If you accepted his offer, he'll want to pray in front of the tree as an alternative to harming the tree.

When he does this, a sapling will grow which you can then take back to Danica to complete the quest.

As others have mentionned, escort Maurice Jondrelle to the Eldergleam Sanctuary. Once at the tree's root, let him pray and return the tree sapling to the Temple of Kynareth.

However be aware that if you use Maurice's suggestion you'll end up with a small tree clipping through the larger, old tree. I'm surprised nobody ever mentions this bug:

This wasn't fixed as of patch 1.3.

After applying a fix through the console (using the markfordelete command), you get this little tree:

So keep this in mind, the "non violent" approach has no benefits that I can think of, and gives you a much less exciting Gildergreen tree, while also giving you less opportunity for skill upgrades through the Spriggan fight.

Who knows... maybe it takes a lot longer and then eventually it will become the bigger version that can be seen on the loading screen?

There is. Return to Whiterun (where you picked up the quest, after acquiring Nettlebane) and some guy will approach you and ask you to escort him to the Eldergleam. Once in Eldergleam, he will basically stop you from doing it and will give you an alternate solution.

Once there, you have to start hacking away at the roots with Nettlebane equipped and after a few chops Maurice will stop you and offer to show you a better way.