unsetting the first element
is easy if you use array_shift. array_shift will pop the first element off and reindex your array for you.
array_shift($quotes_array[0]['methods']);
if you need to unset something in the middle
(such as [ 2 ] out of [ 3 ]) you could use array_values. First unset the element you want to remove, then reindex them with array_values.
unset($quotes_array[0]['methods'][2]);
array_values($quotes_array[0]['methods']);
if you wanted to remove the last element
of an array you could use array_pop. array_pop will pop the last element off the array. There is no need to reindex in this situation.
array_pop($quotes_array[0]['methods']);