When Lord Krishna has chastised Kaliya Naag and ordered him to leave river Yamuna, the Kaliya has requested to free him from the fear of Garuda. Then Shri Krishna has told him that Garuda will not attack on you because you have my footprints on your head.
This incident is mentioned in Srimad Bhagavatam as:
SB 10.16.63 — Out of fear of Garuḍa, you left Ramaṇaka Island and came to take shelter of this lake. But because you are now marked with My footprints, Garuḍa will no longer try to eat you.
And in Harivamsa as:
When you live in the ocean, seeing the marks of my feet on your hoods, garuDa, the enemy of serpents will never harm you.
And as we know that Shri Krishna is the incarnation of Lord Vishnu and Garuda is the mount of Lord Vishnu and Shesha is the bed of Lord Vishnu.
So, in this way Garuda cannot kill or fight with Shesha.
garuDa is the enemy of snakes but this snake list excludes shesha.
mahAbharat Adi parva chapter 36 provides details about relationship between garuDa and shesha :
shesha replied, 'My uterine brothers are all of wicked hearts. I do not desire to live amongst them. Let this be sanctioned by thee. Like enemies they are always jealous of one another. I am, therefore, engaged in ascetic devotions. I will not see them even. They never show any kindness for vinatA and her son. Indeed, vinatA's son capable of ranging through the skies, is another brother of ours. They always envy him.
And the illustrious Grandsire, the best of the immortals, then gave unto Anant the bird of fair feathers, viz., the son of vinataA for ananta's help.'"