The direct transfer of assets from the custodian of the Traditional IRA to the different custodian of the Roth IRA is not discussed at all in the Traditional IRA chapter of Publication 590a but the chapter does say that the IRA owner can take a distribution from a Traditional IRA and within 60 days, roll it over into a Roth IRA, and not have to pay the 10% penalty for taking an early distribution from the Traditional IRA (assuming, of course, that the IRA owner is young enough that the early withdrawal penalty is applicable). Whatever amount came out of the IRA but didn't get rolled over properly in timely fashion (e.g. because taxes were withheld from the distribution and the IRA owner simply sent the amount of the distribution check to the Roth IRA custodian) is subject to the 10% penalty for early distributions. Income tax is certainly due on the total amount of the distribution, less that part of the distribution that is a return of nondeductible post-tax contributions, if any, to the Traditional IRA.
It is also worth pointing out that 60 days is meant literally -- it is not the same as two months as so many people think -- and that the 60 day clock starts from the date that the transmitting custodian records the withdrawal on its books. The deposit into the Roth IRA must occur within 60 days from this date, and the postmark on the envelope in which the check was sent or the date that it was received by the Roth IRA custodian are irrelevant; what matters is the date that the Roth IRA custodian records the money as having been deposited into the Roth IRA.
As @dave-thompson_085 points out, if the question that the OP asked is rephrased as
Is a direct transfer from a Traditional IRA held by one custodian to a Roth IRA with a different custodian prohibited by law? or is it just a rule that is almost universally followed by IRA custodians?
the answer is "Yes, they are permitted" as stated explicitly on page 45 of the Roth IRA chapter of Pub590a (2016 edition; the 2017 tax year edition has not been released as yet) that deals with converting to a Roth IRA. But, there can be good reasons why Traditional IRA Agreements (they do differ somewhat from custodian to custodian) might well not allow requests from the Traditional IRA owner to transfer assets to a Roth IRA with a different custodian. Pub590a says
You can direct the trustee of the traditional IRA to transfer an amount from the traditional IRA to the trustee of the Roth IRA.
which is different from the way that most such transfers might work in practice, where it is the recipient custodian who contacts the Traditional IRA custodian saying something to the effect that ""Our client has a Roth IRA with us/is establishing a Roth IRA with us, and has authorized us to collect the IRA money that you are holding and put it in the Roth IRA".
Does this request from the Roth IRA custodian count as a direct instruction from the IRA owner to the Traditional IRA custodian to do a trustee-to-trustee transfer of funds? I don't know. A literal reading of Pub590a suggests that it might not count as a direct instruction. With a direct instruction as envisioned in the highlighted quote above, the Traditional IRA custodian has no assurances that the recipient Roth IRA exists, or that the recipient custodian even knows about the matter or is expecting the funds. Overall, a can of worms that is best left unopened by prohibiting Traditional to Roth conversions via direct transfer in the Traditional IRA Agreement; let the owner request a distribution and put the onus on the owner to get the entire distribution to the Roth IRA custodian in timely fashion, and be responsible for taxes on what didn't get converted, whether inadvertently or advertently. I note, for example, that Vanguard's website has no forms for requesting a transfer of Vanguard IRA assets to a different custodian (not even for a Traditional to Traditional or Roth to Roth transfer where tax is not an issue) whereas forms for transferring IRA assets to Vanguard can even be completed online. So, either Vanguard's Traditional IRA Agreement (ditto Roth IRA Agreement) doesn't allow direct Vanguard-initiated transfers of IRA assets to another custodian (but will accept requests initiated by the recipient custodian), or the IRA owner has to hassle with Vanguard's Customer Service to demand that such a transfer be done because it is legal and not explicitly forbidden in the IRA agreement.
If the money is staying in house, being transferred from Traditional to Roth, it is guaranteed that the conversion will meet the 60-day rule. Similarly, Traditional to Traditional or Roth to Roth transfers of IRA money from one custodian to another are easy to handle via direct transfer from one custodian to another, but once again, these might need to be initiated by the recipient custodian. Thus, it seems to me that the OP most likely has two options: (i) do an in-house Traditional to Roth conversion at Fidelity and then transfer the Roth IRA to Merrill Edge (can be done as a direct transfer initiated by Merrill Edge, not Fidelity) or (ii) do a Traditional to Traditional transfer from Fidelity to Merrill Edge as a direct transfer (also initiated by Merrill Edge), and then do a Traditional to Roth conversion at Merrill Edge. (ii) is the option that the Merrill Edge customer service is recommending (because it gets the money into Merrill Edge's hands earlier without any hassles of conversions). In either case (ii) or (i), the transfer to Merrill Edge is best initiated by Merrill Edge rather than hassling with Fidelity.