Not automatically. The Gold Card process seeks permanent residence through EB-1 or EB-2, while TN requires temporary intent, so an immigrant filing may affect the TN assessment even if the official Gold Card terms permit parallel processing.
Sequence the two processes only after reading how the program treats other status
The TN application is decided at the port of entry or through Form I-129 under TN's own conditions, including temporary intent. The current Gold Card process uses Form I-140G and seeks lawful permanent residence through an EB-1 or EB-2 classification, so a filing by the nurse or a close family member may be relevant when an officer assesses whether the TN stay is genuinely temporary.
It is not necessarily an automatic bar, but it requires a case-specific intent analysis in addition to checking whether the official Gold Card terms permit the two processes to run together. Resolve both questions before the nurse travels or the couple pays a nonrefundable processing fee or government gift.