I am very definite that it is not love, or at least the love that the OP is hinting at; more along what
Axle broached. In such a setting family members feel responsible in looking after one another, ascertaining that they not go beyond the set standards belonging to their kin, and most respect that through some sort of bond each member shares. Fi was obligated to guiding Link, and she went the extra mile to be sure nothing happened to him, especially in the land below Skyloft, unknown territory to him. This tells me that she was being protective and preventive of harm to Link, though it was his quest...strangely a motherly role comes to mind, albeit that is not the most accurate comparison to Fi's duty.
But that was mostly Fi's perspective during the quest. In the end, as the need of her came to a close, she started to sensing a change in her nature. If the name of songs have anything to do with it, to wit "Fi's Gratitude", then she was grateful of having led Link to the completion of his preordained destiny. Seeing her own task complete she was content to rest. Otherwise we can go with the word she actually
used to express herself:
While many posters before me joked that she'd calculated this feeling, I believe she truly experienced it as we do. For the time that she served her purpose, Fi enjoyed having Link as her Master, yet not out of love, but an affinity which had grown between them.