The Sword Coast Adventurer's Guide states on page 15 that Shieldmeet is a Quadrennial holiday between Flamerule and Eleasis. This is also the case in accordance to the wiki. The Dungeon Master's Guide however states Shieldmeet to be between Hammer and Alturiak on page 33. I am not sure what source(s) is more accurate.
The Sword Coast Adventurer's Guide is Correct
When in doubt, refer to more recent materials first.
At least, that's the rule I'd use in this case. Errata is constantly being discovered in the earlier materials of D&D 5e. So it may be a better idea to refer to the most recently released content (In this case, the Sword Coast Adventurer's Guide) as opposed to the DMG.
However, since errata is so common in this edition, there are two possibilities as to what happened:
- The SCAG was erroneously printed with a later representation of Shieldmeet.
- The DMG was erroneously printed with an earlier representation of Shieldmeet.
There must be some way to get more Forgotten Realms information to cross-compare...
But wait! We do! Forgotten Realms is a long-standing world in the D&D universe, as such this question actually should not be restricted to just 5e. If we refer to the D&D 3e Forgotten Realms Campaign Setting we see the following:
(Forgotten Realms Campaign Setting 3rd Edition p. 77 : Shieldmeet) Shieldmeet is added to the Faerunian calendar as a "leap day" immediately following Midsummer night.
So there you have it. The Sword Coast Adventurer's Guide fixed the errata within the DMG. So use it as the basis for when Shieldmeet is.
Fun fact: Along with the erroneous printing of when Shieldmeet is. Eleasis is also misspelled in the DMG as Eleasias