Simple Summary: For sustainable meat production, beef farmers must make the best use of grass and roughage while limiting the carbon footprint of their herds. The genetic improvement in feed efficiency and enteric methane production of replacement heifers is possible if the recorded phenotypes are available. Intuitively, the relationship between the two traits should be negative, i.e., favorable, since the energy lost with the methane is not available for heifer metabolism. The measurement of feed efficiency requires several weeks of feed intake recording. The enteric methane emission rate can also be recorded over several weeks. The two traits of 326 beef heifers from two experimental farms were measured simultaneously for 8 to 12 weeks. The correlations between roughage intake, daily gain, and methane were all positive. The enteric methane emission rate was positively related to body weight, daily gain, and dry matter intake. The relationship with feed efficiency was slightly positive, i.e., unfavorable. Therefore, the two traits should be recorded simultaneously to evidence low-emitting and efficient heifers. This study also showed that replacing the feed intake recording with the carbon dioxide emission rate appeared potentially beneficial for selecting these low-emitting and efficient heifers.Abstract: Reducing enteric methane production and improving the feed efficiency of heifers on roughage diets are important selection objectives for sustainable beef production. The objective of the current study was to assess the relationship between different methane production and feed efficiency criteria of beef heifers fed ad libitum roughage diets. A total of 326 Charolais heifers aged 22 months were controlled in two farms and fed either a grass silage (n = 252) or a natural meadow hay (n = 74) diet. Methane (CH 4 ) and carbon dioxide (CO 2 ) emission rates (g/day) were measured with GreenFeed systems. The dry matter intake (DMI), average daily gain (ADG), CH 4 and CO 2 were measured over 8 to 12 weeks. Positive correlations were observed among body weight, DMI, ADG, CH 4 and CO 2 . The residual feed intake (r wg DMI) was not related to CH 4 or residual methane (r wi CH 4 ). It was negatively correlated with methane yield (CH 4 /DMI): R p = −0.87 and −0.83. Residual gain (r w iADG) and ADG/DMI were weakly and positively related to residual methane (r wi CH 4 ): R p = 0.21 on average. The ratio ADG/CO 2 appeared to be a useful proxy of ADG/DMI (R p = 0.64 and 0.97) and CH 4 /CO 2 a proxy of methane yield (R p = 0.24 and 0.33) for selecting low-emitting and efficient heifers.