Is it permissible for a sick woman undergoing treatment, who fears infertility due to her illness and treatment, to supplicate to Allah for offspring, or is this considered an objection to Allah’s decree and trial?

Supplicating to Allah does not contradict contentment with His decree. Combining patience and supplication is مطلوب (required/desired), as Prophets Ibrahim and Zakariya (peace be upon them) did in their prayers for a child. Supplication is the most effective remedy for illness and the closest means to obtain a child. Allah responds to those in dire need. Therefore, a Muslim should increase their supplication with sincerity during times when prayers are answered, and avoid haste or despair.