Montoya, who has a previous conviction for murder, was arrested by the Guardia Civil on Tuesday, December 18, on suspicion of killing Luisa Luelmo.
He was captured after a brief chase on foot after his car was stopped.
Miss Luelmo had travelled almost 300 miles from her home town of Zamora in western Spain to take up her first full-time teaching job.
MURDER: Laura Luelmo’s body shows signs of violence and having suffered sexual assault — https://t.co/m65aEaGyfI pic.twitter.com/OTPtwXH5cS
— RTNNewspaper (@RTNNewspaper) 18 December 2018
She went missing on Wednesday, December 12, and her body was found five days later.
It is thought she may have been killed while running or walking in the hills above El Campillo. Detectives are still searching for her purse and smartphone.
She is believed to have been sexually assaulted and strangled during the attack.
A source within the police investigation told the El Pais newspaper Montoya lived in a house belonging to his father which was directly opposite Miss Luelmo's rented apartment.
She started work at a secondary school in nearby Nevera on December 3 but failed to show up for work nine days later.
Montoya and his twin brother Luciano have long criminal records, according to El Pais, and he had only been released from prison earlier this year after serving 17 years of a sentence for murdering an 82-year-old woman with a machete.
Twittter users have been using the hashtag #LauraSomosTodas (We're All Laura) in the last few days.
DIFUNDAN PLIS🙏🏼🙏🏼🙏🏼#LauraSomosTodas #LauraLuelmo #NiUnaMás #niunamenos pic.twitter.com/buQu9YVYjn
— Naty Sujanani (@natysuja) 18 December 2018
The issue of violence against women is a hot topic in Spain at the moment, following the trial of five men earlier this year in connection with the gang rape of an 18-year-old woman during the San Fermín bull-running festival in Pamplona in 2016.
The men became known as La Manada (The Wolf Pack).
This is the last tweet from Laura Luelmo, was for the International Women's Day, a Illustration she made.
— Mariam Fuentes (@MariamFuentesss) 18 December 2018
She was taken, and murdered, her body has just been found.#ViolenceAgainstWomen https://t.co/fdTAxGToPw
They were eventually acquitted of rape and convicted of a lesser charge.
Miss Luelmo, who was a talented illustrator, designed an image to mark International Women's Day in March and it has been widely shared in the days since she disappeared.
Justicia. https://t.co/n5ZihSRJdM
— Loola Pérez (@DoctoraGlas) 18 December 2018
"They teach you not to go alone in dark places instead of teaching monsters not to be one. THAT's the problem," Miss Luelmo wrote on Twitter in 2015, a message which has been retweeted thousands of times in the last 24 hours.
Schools across Spain observed a minute's silence on Tuesday and several Spanish politicians commented on the case.
"Women have to pay cruel and extremely high prices for our freedom and security," said Spain's Deputy Prime Minister, Carmen Calvo, who is also the Equality Minister.
