Husk, 5 6 bytes

€ḣ¹tẊ-

Try it online!

Get the pairwise differences (Ẋ-), remove the first one (t), and check if this is an ordered sublist one of the prefixes of the input (€ḣ¹).

Of course, it needs to be a prefix of the input (rather than a sublist at any other position), but I think this should be ensured since the input is increasing. Edit: the fact that the input is increasing does not ensure that sublists must be prefixes (consider 1 2 4 with difference sublist 2), so we need to specify this directly, at the cost of 1-byte.

Husk, 5 6 5 bytes

€¹tẊ-

Try it online!

Outputs 1 for truthy (it's a fibonacci-like sequence), any other non-negative integer for falsy (it isn't a fibonacci-like sequence).

Get the pairwise differences (Ẋ-), remove the first one (t), and check if this is an ordered sublist of the input find the index of this sublist in the input (€ḣ¹), or 0 if not present.

Of course, it needs to be a prefix of the input (rather than a sublist at any other position), but I think this should be ensured since the input is increasing. Edit: the fact that the input is increasing does not ensure that sublists must be prefixes (consider 1 2 4 with difference sublist 2), so we check this by requiring that the index of the sublist must be 1.

Husk, 5 bytes

€¹tẊ-

Try it online!

Get the pairwise differences (Ẋ-), remove the first one (t), and check if this is an ordered sublist of the input (€¹).
  Of course, it needs to be a prefix of the input (rather than a sublist at any other position), but I think this should be ensured since the input is increasing.

Husk, 5 6 bytes

€ḣ¹tẊ-

Try it online!

Get the pairwise differences (Ẋ-), remove the first one (t), and check if this is an ordered sublist one of the prefixes of the input (€ḣ¹).

Of course, it needs to be a prefix of the input (rather than a sublist at any other position), but I think this should be ensured since the input is increasing. Edit: the fact that the input is increasing does not ensure that sublists must be prefixes (consider 1 2 4 with difference sublist 2), so we need to specify this directly, at the cost of 1-byte.