It's been a year since your post & I don't know if you've found the information you've been searching for. If not, this is what you can do. Alexandra Gutierrez was born in Mexico. If she became a U.S. citizen and obtained a SS#, then you could write to the SS Administration for a copy of the SS# application which might have her place of birth. Or you could order the SS application of any the children born in the U.S. and those might have the parents place of birth.
I also found a couple of border crossing records which may be your family's.
Alexandria Gutierrez age 3/12
dob a about 1923, Juarez, Chihuahua
year of immigration 21 Sept1923
mother- Eugenia Rosaleo(spelling?)
port of entryEl Paso, Texas, USA
born about 1891
arrived 17Jan 1921 El Paso , Tx
from Conchos, Chihuahua
Birth about 1900, age 22
Arrived 01 Oct 1922/ El Paso, Tx
Birth place Moclavia(?) Coahuila, Mx
Last residence Juarez, Chihuahua, Mx
Accompanied by Alexandra Gutierrez
If you can obtain a death certificate
for Eugenia Gutierrez it might have her
maiden name on it.