Born in Monteverde, Puntarenas, Moya graduated with Deportivo Saprissa's youth setup, and made his senior debuts with the reserves in Segunda División, on 3 February 2010, in a match against Cariari.

Jonathan was loaned out to Santos de Guápiles in early 2012, and made his debut in the Primera División in a 1–0 home victory over Herediano on 15 January.

Although his team lost their match 1–0 for the semifinals against Deportivo Saprissa, they qualified to the last round after winning the second leg with the same score, this one due to the advantage earned by finishing second in the regular season.

Once Angulo was bought on 16 December 2013, the deal established that two member from Saprissa would move to Coronado on loan, so Moya and his team partner Dylan Flores were chosen to close the transfer.

As a result of his scoring performance, Jonathan was awarded in late May as the top scorer, sharing the prize with the Uruguayan Fabrizio Ronchetti and the Argentine Lucas Gómez.

[4] He was presented in press conference on 6 January 2015 –same day of his birthday–, by the financial manager Galeano Luconi, along with the other new players Néstor Monge and Andrés Imperiale.

Moya finally made his official debut for the first squad on January 19, 2015, in a match against AS Puma at the Ricardo Saprissa, playing 36' minutes in the 3–2 victory.

On 27 January 2016, the team announced his departure, because of the lack of opportunities he had to play, and for arriving eleven days late to the trainings after the pause of the half of the season.

Jonathan replaced Juan Bustos Golobio at the 70th minute and gave an assist to David Ramírez, who scored the goal of the 1–0 victory over Universidad de Costa Rica at the "Fello" Meza.

As a result of the new players in the team, who have a chance to be part of the official squad at the 2018 World Cup, Moya was relegated to the bench by the coach Vladimir Quesada.

After three successful seasons in Liga FPD with Alajuelense, on January 11, 2021, the club announced that Moya would join South Korean team FC Anyang on loan for the rest of the year.

[6] Playing in K League 2, the forward registered his third experience outside his native country, as well as the first one in the Asian continent: plus, he became in the process the fourth Costa Rican player to join the South Korean professional football system, after Jeaustin Campos, Elias Aguilar and Marco Ureña.

[8] After representing Costa Rica at under-17 and under-20 levels, Moya made his main squad debut on 14 September 2014, replacing David Ramírez in the dying minutes of a 2–1 win against Guatemala, for the year's Copa Centroamericana.