The spike growth effect must be within the druid grove area; it’s unclear whether it must be entirely within the area
The druid grove spell includes a few potentially helpful lines (emphasis mine):
(…) The area can be as small as a 30-foot cube or as large as a 90-foot cube. (…)
(…) The spell creates the following effects within the area. (…)
(…) A dispel magic cast on the area, if successful, removes only one of the following effects, not the entire area. (…)
(…) You can place your choice of one of the following magical effects within the warded area: (…)
From the first quote we can conclude that “the area” refers to the actual grove itself not the various effects of the spell. Thus we know that the effects (including the additional spell effects) are created within the area. We also know from the fourth quote that these effects are placed within the area. What we do not know is whether “the effect” means the point-of-origin or the entire effect. I could see a GM using either interpretation.
My own personal thoughts
If I tell you to put a book on the table and you have the book teetering off the edge, I will smile, say “really?”, and move the book to be actually on the table.
While the spike growth might be inside the druid grove it also goes outside the grove. It would be meeting a technical requirement because the spell did not include the word “entirely” but I wouldn’t allow it to be placed in such a way.