Unless your new disk is over 2TiB, I strongly advise against attempting an MBR-to-GPT conversion on a Windows boot disk. The reason is that Windows ties its boot method (BIOS vs. EFI) to the disk type, so doing such a conversion requires switching from a BIOS-mode boot to an EFI-mode boot of Windows. If you're sure your firmware supports EFI, then this is at least possible, but it's still a hassle. See this page for details of how to do it. Basically, you need to use gdisk to do an MBR-to-GPT conversion, create a new EFI System Partition (ESP) to hold the EFI boot loader, and set up the EFI boot loader. You'll need to make similar changes in Linux, too, but they can be done a little more easily.
Let's be careful about drives vs. partitions. A drive is an entire physical storage device: a hard disk drive. A drive is something you can hold in your hand.
The storage space on the drive may be divided up into one or more partitions. Partitions are not drives, they are sections of drives. You cannot hold a partition in your hand.
GPT and MBR are drive-level partition table formats that affect the whole hard disk drive. It's how you tell your system what the partitions on the drive are, how big they are, and where they start and end.
The ancient partition table format from the 1980's IBM PC BIOS firmware era is MBR. The modern format from the 2000's Intel [U]EFI firmware era is GPT.
A drive could either have only an MBR table, or it could have only a GPT table, or it could be a hybrid containing both.
One form of hybrid is primarily GPT but has a "Protective MBR" or "PMBR": a fake MBR that tells non-GPT-aware software that the whole disk is allocated to an unknown partition type. This keeps non-GPT-aware software from thinking the disk is unformatted and trying to format it.
Another form of hybrid is a disk that has both a GPT and an MBR, where the formatting/partitioning software has tried to keep them both in sync. So MBR-only software learns about all the same partitions that GPT-only software learns about.
You can make a GPT-only drive into a hybrid, but since this concept is at the whole-disk level and not the individual-partition level, you can't make a single partition be MBR.
Partitions have a different set of formats, such as FAT32, ExFAT, NTFS, HFS+, and others. These are choices that affect individual partitions, not the whole disk drive.
Best Answer
Not cheap - $50 - but Paragon Hard Disk Manager would appear to be able to do that, without data loss - though I'd always recommend you have a backup before attempting anything like that.