No. X can't be 0 for Conflagrate if it has at least one target, since for that spell, X damage is to be divided among "any number of" targets (which "may be creatures, players, or planeswalkers"), and each target must have at least 1 damage assigned to it this way (C.R. 601.2d, 114.4; see also C.R. 107.1c). See also this thread.
EDIT: Correctness edit after comment 3 was posted.
With Core Set 2020, the answer to comment 1 has changed.
The rule update for Core Set 2020 deleted the exception that applied, among other things, to Conflagrate's "any number" (compare C.R. 107.1c and 608.2d in Modern Horizons with those rules in Core Set 2020). Before Core Set 2020, an X of 0 could be chosen for Conflagrate only if it wasn't possible to choose one or more targets for that spell. Once Core Set 2020 is released (and in prereleases), X can be chosen as 0 for Conflagrate (and zero targets chosen for that spell) regardless of whether Conflagrate could have had one or more targets rather than none.
No. X can't be 0 for Conflagrate if it has at least one target, since for that spell, X damage is to be divided among "any number of" targets (which "may be creatures, players, or planeswalkers"), and each target must have at least 1 damage assigned to it this way (C.R. 601.2d, 114.4; see also C.R. 107.1c). See also this thread.
EDIT: Correctness edit after comment 3 was posted.
EDIT (Jul. 9, 2019): See comment 4.
The rule update for Core Set 2020 deleted the exception that applied, among other things, to Conflagrate's "any number" (compare C.R. 107.1c and 608.2d in Modern Horizons with those rules in Core Set 2020). Before Core Set 2020, an X of 0 could be chosen for Conflagrate only if it wasn't possible to choose one or more targets for that spell. Once Core Set 2020 is released (and in prereleases), X can be chosen as 0 for Conflagrate (and zero targets chosen for that spell) regardless of whether Conflagrate could have had one or more targets rather than none.